Poisson's Equation and Laplace's Equation in Electrostatics

Introduction

• Topic: Poisson’s Equation and Laplace's Equation for electrostatic fields
• Goals: Review concepts of Poisson and Laplace's equations, and solve an example problem

Key Concepts

Poisson's Equation

• Type: Non-homogeneous differential equation
• Derivation:
  • Starts from Gauss's Law: Divergence of electric flux density field D is equal to volumetric charge density ρᵥ
  • Constitutive relation in free space links D with electric field E
  • Substituting this relation into Gauss's law gives: ∇ • E (Divergence of E)
  • Electric field E related to potential V through gradient: E = -∇V
  • Results in second-order differential equation: ∇ • (∇V) = -ρᵥ / ε
    • Divergence of Gradient: Cartesian coordinates: ∇²V = -ρᵥ / ε
    • Laplacian Operator: ∇²
• Forms in Different Coordinates:
  • Cartesian: ∇²V = ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z²
  • Cylindrical: More complex form
  • Spherical: Even longer form

Laplace's Equation

• Type: Homogeneous form of Poisson's equation
• Special Case: Right-hand side of Poisson’s Equation is zero: ∇²V = 0
• Application: Holds outside the source region where the volumetric charge density ρᵥ = 0
• Uses: Employed where observation point is outside of charge distributions (volumetric, surface, line, point charges), simplifying to solve as boundary value problem
• General Application: Applies to various charge distributions (volumetric, surface, line, point charges)

Example Problem

Problem Statement

• System: Parallel plate capacitor
  • Plates at y = a (Voltage V(a)) and y = b (Voltage V(b))
• Objective: Find voltage and electric field in the region between the plates

Solution Steps

1. Apply Laplace's Equation: Works in the region between the plates ∇²V = 0
2. Work in Cartesian Coordinates: Simplifies as variation is only in Y-direction
   • Expression: ∂²V/∂y² = 0
3. Integrate: Solve first-order and second-order integrals
   • First integration: dV/dy = A
   • Second integration: V = Ay + B
4. Determine Constants:
   • Boundary conditions: V(a) = Vᵃ and V(b) = Vᵇ
   • Solve for A and B using boundary conditions
     • Set V(a) = 0 (reference) and V(b) = V₀
     • Find: A = V₀ / (b - a), B = - a * (V₀ / (b - a))
   • Final voltage expression: V(y) = (V₀ / (b - a)) * (y - a)

Electric Field Calculation

• Relation: E = -∇V
  • Only Y-component matters
  • Electric Field: E = - (V₀ / (b - a)) ŷ
  • Result: Uniform electric field pointing in negative Y direction

Conclusion

• Poisson vs Laplace: Laplace's equation simplifies when in source-free region
• Methods: Alternative way to find electric field and potential
  • Coulomb’s/Gauss's Law
  • Laplace's equation for electrostatics boundary value problems